Symbolic dynamics of noisy chaos

Crutchfield, James P.; Packard, Norman H.

doi:10.1016/0167-2789(83)90127-6

Cited by 174 publications

(135 citation statements)

References 34 publications

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“…Among them, the time delay embedded reconstruction [1] provides a general framework for the estimation of invariant quantities (in terms of smooth transformations of the state space of the attractor) of the original system, such as attractor dimensions, Lyapunov exponents and entropies [2][3] [4]. Information theoretic measures of structural complexity include effective measure complexity (EMC) [5], excess entropy [6], and predictive information [7] [8].…”

Section: Introductionmentioning

confidence: 99%

Multivariate multiscale entropy: A tool for complexity analysis of multichannel data

2011

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This work generalizes the recently introduced univariate multiscale entropy (MSE) analysis to the multivariate case. This is achieved by introducing multivariate sample entropy (MSampEn) in a rigorous way, in order to account for both within-and cross-channel dependencies in multiple data channels, and by evaluating it over multiple temporal scales. The multivariate MSE (MMSE) method is shown to provide an assessment of the underlying dynamical richness of multichannel observations, and more degrees of freedom in the analysis than standard MSE. The benefits of the proposed approach are illustrated by simulations on complexity analysis of multivariate stochastic processes and on real world multichannel physiological and environmental data.

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Section: Introductionmentioning

confidence: 99%

Multivariate multiscale entropy: A tool for complexity analysis of multichannel data

2011

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“…They are actually known for only a few synthetic examples such as the torus map, the standard map or the Henon map [36][37][38]. Therefore, algorithms have been suggested to estimate them from experimental time series [39][40][41][42].…”

Section: Empirical Constructionmentioning

confidence: 99%

Identifying mental states from neural states under mental constraints

Atmanspacher

2011

Interface Focus.

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This article emphasizes how the recently proposed interlevel relation of contextual emergence for scientific descriptions combines 'bottom-up' and 'top-down' kinds of influence. As emergent behaviour arises from features pertaining to lower level descriptions, there is a clear bottom-up component. But, in general, this is not sufficient to formulate interlevel relations stringently. Higher level contextual constraints are needed to equip the lower level description with those details appropriate for the desired higher level description to emerge. These contextual constraints yield some kind of 'downward confinement', a term that avoids the sometimes misleading notion of 'downward causation'. This will be illustrated for the example of relations between (lower level) neural states and (higher level) mental states.

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“…If E is a generating partition, then all refinements of E are also generating partitions. Furthermore, if E is generating, then the sigma algebra generated by all refinements of E coincides with the sigma algebra of all measurable subsets of M [28][29][30].…”

Section: Symbolic Dynamicsmentioning

confidence: 99%

“…where the supremum is taken over all measurable partitions E and where [28][29][30]. Systems with a positive KS entropy are called K systems.…”

Section: Symbolic Dynamicsmentioning

confidence: 99%

Chaos for Liouville probability densities

Schack

Caves

1996

Phys. Rev. E

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Using the method of symbolic dynamics, we show that a large class of classical chaotic maps exhibit exponential hypersensitivity to perturbation, i.e., a rapid increase with time of the information needed to describe the perturbed time evolution of the Liouville density, the information attaining values that are exponentially larger than the entropy increase that results from averaging over the perturbation. The exponential rate of growth of the ratio of information to entropy is given by the Kolmogorov-Sinai entropy of the map. These findings generalize and extend results obtained for the baker's map [R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992)].

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Symbolic dynamics of noisy chaos

Cited by 174 publications

References 34 publications

Multivariate multiscale entropy: A tool for complexity analysis of multichannel data

Multivariate multiscale entropy: A tool for complexity analysis of multichannel data

Identifying mental states from neural states under mental constraints

Chaos for Liouville probability densities

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